The Galactic rotation period at the Sun's Galactocentric radius is about 230 million years (with a five percent uncertainty) and the Sun, as stated in user8's answer is 4.57 billion years old - giving $\sim 20\pm 1$ orbits.

However, the idea of a Galactic year is misleading. For instance, the Milky Way rotation curve is quite flat between 1 and 10 kpc from the centre (see picture), so the "year" also varies by a factor of 10 over this range, with the year being much shorter in the inner part of the Galaxy.

Therefore to know how many rotation periods the Sun has executed requires us to know at what Galactocentric radius it has spent its life. It is an ongoing debate as to whether the Sun has migrated to its position from outside or inside its current radius, or whether it has been where it is all along. As a result, how many "Galactic years" our Sun has experienced is still quite uncertain.